# Definition:Small Class

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## Definition

Let $A$ denote an arbitrary class.

Then $A$ is said to be **small** if and only if:

- $\exists x: x = A$

where $=$ denotes class equality and $x$ is a set variable.

That is, a class is **small** if and only if it is equal to some set variable.

To denote that a class $A$ is **small**, the notation $\mathcal M \left({A}\right)$ may be used.

Thus:

- $\mathcal M \left({A}\right) \iff \exists x: x = A$

## Remark

**Small classes** intuitively correspond to sets.

They are, however, formally distinct from set variables.

Namely, statements about set variables can be done inside the language of set theory.

On the contrary, statements about classes first need to be "rewritten" as explained in the definition of class in Zermelo-Fraenkel set theory.

By Set is Small Class, all set variables can be considered **small classes**.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.10$